It is well known that if \(\Theta \subseteq \lbrace a\in \mathbb {R}^d|\Vert a\Vert _2\le 1\rbrace \) , then there is \(1/T\) -net of \(\Theta \) , \(\Theta ^{\prime }\) , such that \(|\Theta ^{\prime }|\le (6T)^{d}, \Theta ^{\prime }\subseteq \Theta \) . This directly implies that \(R_T=O(d\sqrt{T\log (T)})\)  with probability at least \(1-1/T\) . This improves a factor of \(\sqrt{\log T}\log \log T\)  over [1]} and a factor of \(\sqrt{\log T}\)  over [2]}.
